Problem: All of the 3rd grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$2.00$ each for students, and the group paid $$35.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$17.00$ each for teachers and $$9.50$ each for students, and the group paid $$119.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2y = 35}$ ${17x+9.5y = 119.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-4y = -70}$ ${17x+9.5y = 119.5}$ Add the top and bottom equations together. $ 5.5y = 49.5 $ $ y = \dfrac{49.5}{5.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8.5x+2y = 35}$ to find $x$ ${8.5x + 2}{(9)}{= 35}$ $8.5x+18 = 35$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {17x+9.5y = 119.5}$ and get the same answer for $x$ ${17x + 9.5}{(9)}{= 119.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.